Moduli of A-Infinity Structures and Related Topics

该项目是在代数几何领域与弦理论的一些连接。代数几何是数学的一个分支，研究由多项式方程和相关数学结构定义的几何对象。经典地，人们将这样的几何对象（称为代数簇）与它们上的代数函数的集合相关联，该集合形成交换环（即，函数可以相加和相乘）。PI研究与代数簇相关的更复杂的代数结构，例如A-无穷代数和层的导出范畴。在A-无穷代数中，除了两个元素的乘积之外，还有将n元组元素相乘的更高运算。这些结构在福谷的作品中以一种非常不同的方式独立出现。这项研究的部分动机是为了匹配A-无穷代数的不同构造。PI还打算应用A-无穷结构来解决代数几何中的一些问题。该项目将主要集中在以下两个主题：A-无穷结构的模空间和与代数曲线相关的模空间。在项目的第一部分，目标是构建模空间（即，参数空间）的A-无限结构相关的各种模空间的几何对象（有时涉及非交换几何）。特别是，PI计划构建与代数曲面相关的A-无限代数的模空间，非交换射影线的双覆盖，曲线上的非交换阶，以及与双有理变换相关的A-无限模的模空间。第二部分研究了具有非特殊因子的曲线模空间的复曲面GIT图，以及向量丛曲线模空间及其非交换模空间的GIT图。此外，PI计划研究具有非特殊因子的曲线的模空间上的自然向量场，并使用它们来产生没有标记点的曲线的模空间上的显式有理函数。

The project is in the field of algebraic geometry with some connections to string theory. Algebraic geometry is a branch of mathematics studying geometric objects defined by polynomial equations and related mathematical structures. Classically one associates with such geometric objects (called algebraic varieties) the set of algebraic functions on them which forms a commutative ring (i.e., functions can be added and multiplied). The PI studies much more sophisticated algebraic structures associated with algebraic varieties, such as A-infinity algebras and derived categories of sheaves. In an A-infinity algebra, in addition to taking product of two elements, one has higher operations of multiplying together n-tuples of elements. These structures appeared independently in a very different way in works of Fukaya. The research is partly motivated by the desire to match different constructions of A-infinity algebras. The PI also intends to apply A-infinity structures to solve some problems in algebraic geometry.The project will focus mostly on the following two topics: moduli spaces of A-infinity structures and moduli spaces related to algebraic curves. In the first part of the project the goal is to construct moduli spaces (i.e., parameter spaces) of A-infinity structures related to various moduli spaces of geometric objects (sometimes involving non-commutative geometry). In particular, the PI plans to construct moduli spaces of A-infinity algebras related to algebraic surfaces, double covers of non-commutative projective lines, noncommutative orders over curves, as well as moduli spaces of A-infinity modules related to birational transformations. The second part of the project is devoted to studying toric GIT picture for moduli spaces of curves with nonspecial divisors, as well as moduli spaces of curves with vector bundles on them and their non-commutative analogs. Also, the PI plans to study natural vector fields on the moduli spaces of curves with nonspecial divisors and use them to produce explicit rational functions on the moduli spaces of curves with no marked points.